Let $A$ be bounded self adjoint operator. Then, at least one or both of $\|A\|$, $-\|A\|$ is in spectrum $\sigma(A)$. I can not prove this.
I'd appreciate it if you could help.
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This is fairly clear from the Spectral Theorem. Maybe that counts as too easy and we want a solution from first principles... – David C. Ullrich Aug 26 '19 at 12:10
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Thanks! I find the theorem, a necessary and sufficient condition for that self adjoint operator $T$ is bounded is spectral family $E(\lambda)$ is bounded, and then $|T|=\max{|\min E|, |\max E|}$. – sate Aug 26 '19 at 20:27